A congruence \(\vartheta \in \operatorname{Con}(\mathbf {A})\) 
is said to be completely meet irreducible if
for any family \(\lbrace \vartheta _j:j\in J\rbrace \) 
of congruences of \(\mathbf {A}\) ,
\(\vartheta =\bigwedge _{j\in J}\vartheta _j\) 
implies that \(\vartheta =\vartheta _j\)  for some \(j\in J\) .
If \(\vartheta \)  is completely meet irreducible
and \(\vartheta \ne 1\) , then \(\vartheta \) 
has a unique upper cover
\(\vartheta ^+\)  in \(\operatorname{Con}(\mathbf {A})\) , that is, there is a unique
congruence \(\vartheta ^+\in \operatorname{Con}(\mathbf {A})\) 
such that \(\vartheta <\vartheta ^+\)  and
\({[\![}\, {\vartheta },{\vartheta ^+}\,{]\!]}=\lbrace \vartheta ,\vartheta ^+\rbrace \) .
We will use the notation \(\operatorname{Irr}(\mathbf {A})\)  for the
set of all completely
meet irreducible congruences of \(\mathbf {A}\) , excluding 1.
An algebra \(\mathbf {A}\)  is said to be
subdirectly irreducible if
\(0\,(\ne 1)\)  is completely meet irreducible in \(\operatorname{Con}(\mathbf {A})\) .
In this case
the unique minimal non-trivial congruence \(0^+\)  is
called the monolith of \(\mathbf {A}\) .
It follows from the second isomorphism theorem
([1]})
that
the subdirectly irreducible quotients of \(\mathbf {A}\) 
are exactly the algebras \(\mathbf {A}/\sigma \) , \(\sigma \in \operatorname{Irr}(\mathbf {A})\) .
